Problem 29 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ \frac{3}{8} x-\frac{1}{6}=-\frac{5}{8} x-\frac{2}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{2}\).

Step 1: Clear the Fractions

To make calculations easier, eliminate fractions by multiplying every term by the least common multiple (LCM) of the denominators. The denominators are 8, 6, and 3. The LCM of 8, 6, and 3 is 24. Multiply every term in the equation by 24:\[ 24 \times \left( \frac{3}{8} x \right) - 24 \times \left( \frac{1}{6} \right) = 24 \times \left( -\frac{5}{8} x \right) - 24 \times \left( \frac{2}{3} \right) \]This simplifies to:\[ 9x - 4 = -15x - 16 \]

Step 2: Combine Like Terms

To solve for \(x\), first get all the \(x\)-terms on one side and all the constant terms on the other side. Add \(15x\) to both sides to move the \(x\)-terms to one side:\[ 9x + 15x - 4 = -15x + 15x - 16 \]This simplifies to:\[ 24x - 4 = -16 \]

Step 3: Isolate the Variable

Next, isolate \(x\) by getting rid of the constant term on the left side. Add 4 to both sides:\[ 24x - 4 + 4 = -16 + 4 \]Simplifying gives:\[ 24x = -12 \]

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides by 24:\[ x = \frac{-12}{24} \]This simplifies to:\[ x = -\frac{1}{2} \]

Step 5: Check the Solution

Substitute \(x = -\frac{1}{2}\) back into the original equation to verify the solution:\[ \frac{3}{8} \left( -\frac{1}{2} \right) - \frac{1}{6} = -\frac{5}{8} \left( -\frac{1}{2} \right) - \frac{2}{3} \]Simplify both sides:Left side: \(-\frac{3}{16} - \frac{1}{6}\)Right side: \(\frac{5}{16} - \frac{2}{3}\)Convert to a common denominator (48):Left: \(-\frac{9}{48} - \frac{8}{48} = -\frac{17}{48}\)Right: \(\frac{15}{48} - \frac{32}{48} = -\frac{17}{48}\)Both sides are equal, confirming \(x = -\frac{1}{2}\) is indeed the solution.

Key Concepts

Fractions and Mixed Numbers

Solving equations with fractions and mixed numbers can initially seem daunting. However, understanding their properties makes calculations more approachable.
Fractions represent parts of a whole and have two components: the numerator and the denominator. Mixed numbers are a combination of a whole number and a fraction. When solving equations, it's useful to convert mixed numbers into improper fractions. This makes it easier to manage arithmetic operations.
When fractions appear in equations, consider clearing them by finding a least common multiple (LCM) of the denominators, transforming the equation into one with whole numbers. This step simplifies further solution processes. As an example, in the original problem, fractions appear as coefficients and constants. Simplifying fractions early relieves complexity later on.

Least Common Multiple

The least common multiple (LCM) is a key tool in solving linear equations involving fractions. It is the smallest multiple that is exactly divisible by each of the numbers you're working with. The LCM allows you to clear fractions from an equation, transforming it from fractional coefficients to whole numbers.
For example, in the exercise with the equation \[ \frac{3}{8} x - \frac{1}{6} = -\frac{5}{8} x - \frac{2}{3} \], the LCM of 8, 6, and 3 is 24.
Multiplying every term by 24 produces a new, cleaner equation: \[ 9x - 4 = -15x - 16 \].
This transformation makes it easier to handle operations like combining like terms and isolating the variable. Understanding LCM helps you convert a complex equation into a format that is easier to solve.

Checking Solutions

Checking solutions is an integral part of solving equations. It's essential to confirm that the proposed solution is correct and satisfies the original equation.
To verify, substitute the solution back into the initial equation and ensure both sides balance. For instance, in our problem, with the solution \( x = -\frac{1}{2} \), substitution involves:

• Calculate the left side: \( \frac{3}{8} \left( -\frac{1}{2} \right) - \frac{1}{6} \) simplifies to \( -\frac{17}{48} \).
• Calculate the right side: \( -\frac{5}{8} \left( -\frac{1}{2} \right) - \frac{2}{3} \) also simplifies to \( -\frac{17}{48} \).

Since both sides are equal, the solution \( x = -\frac{1}{2} \) is indeed correct.
Checking solutions not only proves accuracy but also boosts confidence in your problem-solving skills.